# T  I  M     P  E  R  U  T  Z

Tim Perutz
Assistant Professor, Department of Mathematics,
University of Texas at Austin   Research interests:

In a little more detail:

Symplectic topology

Symplectic manifolds arise both from mathematical physics (as phase-spaces for hamiltonian systems) and from algebraic geometry (as smooth quasi-projective varieties over the complex numbers). Symplectic manifolds can be approached from many angles, and in connection with many mathematical fields (integrable systems, representation theory, geometric quantization, Poisson geometry, the algebraic topology of moment maps, …). But symplectic topology refers specifically to the study of global aspects of symplectic manifolds. Its core questions ask about when a smooth manifold admits a symplectic structure, to what extent that structure is unique, and to what extent its symplectic symmetries coincide with those of the underlying smooth manifold.

The word “symplectic” was coined by Hermann Weyl in his book “The Classical Groups”, and has an appropriately classical etymology. Weyl, studying what we call the linear symplectic group, had thought of using the adjective “complex”, but that clashed with complex numbers, or the adjective “Abelian” in honor of Abel, perhaps the first to study this group, but that clashed with the term for commutative groups. So he substituted the Greek equivalent “symplectic” for the Latin “complex”.

Pseudo-holomorphic curve techniques

The tangent spaces of a symplectic manifold can be made into complex vector spaces (this involves a choice J for how i will act on tangent vectors, but the choice is in some ways inessential). Symplectic topologists probe symplectic manifolds X using pseudo-holomorphic curves, i.e., maps u: S → X from a Riemann surface S into X whose derivative Du is complex-linear with respect to J.

One thing you can do with pseudo-holomorphic curves is “count” them (the scare quotes are there because these counts are really weighted and signed, or worse). Such counts are called Gromov-Witten invariants. For instance, there are 2875 degree 1 pseudo-holomorphic curves of degree 1 from the 2-sphere to a quintic 3-fold Q in complex projective 4-space, and the same is true of any symplectic manifold which is symplectically equivalent to Q.

Another thing you can do with pseudo-holomorphic curves is use them to set up Floer cohomology HF(L,L') for pairs (L,L') of Lagrangian submanifolds in a symplectic manifold. This is a vector space over a base field k, the cohomology of a cochain complex CF(L,L').

We can use pseudo-holomorphic curves in yet another way, studying pseudo-holomorphic polygons with Lagrangian boundary conditions. These get turned into algebra by saying that collectively, the Lagrangian submanifolds of X form the objects, and the Floer cochain complexes CF(L,L') form the morphism-spaces, in an A category, a structure in which composition is not strictly associative, but is associative up to a coherent series of homotopies. This structure is called the Fukaya category and may be denoted by F(X).

To “compute” F(X) in full would involve describing all the Lagrangian submanifolds, their Floer cochain complexes, and their A-structure. That is wholly impractical.

In recent years, symplectic topologists have increasingly come to appreciate an insight from homological algebra, something that was long understood by algebraic geometers studying coherent sheaves on algebraic varieties. The insight is that one should not attempt to study F(X) up to derived Morita equivalence, meaning, up to equivalence of an associated category Mod F(X) of A-modules. The derived Morita equivalence class of F(X) is, increasingly, a tractable invariant. It plays a central role in my research.

Mirror symmetry

The first thing to say about “Mirror symmetry” is that it name is metaphorical: while there is a mathematical theory that grows from the idea of reflection in a mirror as a symmetry, this is not it. Instead, it's about looking-glass worlds like Lewis Carroll's, in which things one knows about reappear, but transformed into quite different forms.

Mirror symmetry instead refers to the mathematical explanation of an idea from string theory. The first claim (stated somewhat inaccurately, like everything else in this explanation) is that Calabi-Yau manifolds (which in string theory are candidates for mathematical models of the universe) come in “mirror” pairs, X and Y. On X one consider both symplectic and complex algebraic geometry, and likewise on Y. The second claim is that, in numerous, specific, and extraordinary ways, if one looks just at the symplectic topology of X, one finds that it is identical to the complex algebraic geometry of Y.

Core homological mirror symmetry project

There are three superficially quite different formulations of mirror symmetry between X and Y:

• geometric (T-duality);
• homological mirror symmetry (HMS); and
• Hodge-theoretic.

Geometric mirror symmetry explain how to construct mirror pairs (X,Y). For example, there is a now-classical construction of the mirror the quintic 3-fold, as well as some rather general modern methods (Kontsevich-Soibelman, Gross-Siebert, Fukaya-Abouzaid, …).

Kontsevich's homological mirror symmetry philosophy proposes that, for mirror pairs, there should be a derived Morita equivalence of the Fukaya category F(X) and the derived category of coherent sheaves DCoh(Y).

Hodge-theoretic mirror symmetry says that the A-model (semi-infinite) variation of Hodge structures of X, which is based on the cohomology H*(X) and on genus 0 pseudo-holomorphic curves, is equivalent to the B-model (semi-infinite) variation of Hodge structures of Y. Though abstract, Hodge-theoretic mirror symmetry has very concrete implications for Gromov-Witten invariants (beginning with the 2875 degree 1 curves on the quintic 3-folds).

A widely-shared expectation (loose conjecture) is that geometric ⇒ homological ⇒ Hodge-theoretic mirror symmetry.

In a research project with S. Ganatra and N. Sheridan, we prove that homological mirror pairs are Hodge-theoretic mirror pairs. More precisely, we formulate the notion of core homological mirror symmetry, which asks for an equivalence between some full subcategory A of F(X) and some full subcategory B of DCoh(Y), where the inclusion of B into DCoh(Y) is a derived Morita invariance. We prove that (under fairly mild assumptions)

core homological ⇒ homological ⇒ Hodge-theoretic mirror symmetry.

We have not proved that any large class of geometric mirror pairs obey HMS (equivalently, core HMS), but we overcome one major difficulty in doing so, by showing that it suffices to prove core HMS. Thus you may work with any convenient and reasonably large collection of Lagrangian submanifolds, and not with arbitrary Lagrangians that you have no control over.

Low-dimensional topology

Explanation in progress …

Email: perutz AT math.utexas.edu

Updated: August 2015 (update ongoing)